The long term objective of this research project is to develop an effective computational model analyzing and assessing fluid mechanical parameters and characteristics important to the application of intra-aortic balloon pumping. The mathematical and computational techniques to be developed will extend the dimension and application of digital computer in the development and operation of cardiac assist devices. The potential of reducing or replacing some of the in-vitro and in-vivo experiments for this clinical cardiovascular problem relies on the feasibility of computational simulation of nonlinear pulsating blood flow interacting with the intra-aortic balloon counter-pulsation. The blood flows can be characterized by unsteady flow with moving boundaries. The Navier-Stokes equations will be solved with the dynamic equation of a balloon. Also important to the unsteady flow processes is the effect of distensibility of vessel wall on the hemodynamic characteristics. The complexity of the computational flow simulation will be tractable when the time-dependent domain of the flow field is mapped onto a fixed region in a new coordinate system. The time-dependent pressure drop between the inlet and the outlet and the movements of the balloon and the aorta are grouped as the dimensionless parameters in the Navier-Stokes equations. In short, the key point of the proposed approach is to transfer and cast the time-dependent boundary conditions--geometrical, kinematical, and dynamical--into the equations of motion. Some insight into the blood flow processes around this cardiac assist device will be obtained. The magnitudes of shear stresses, flow patterns, and flow separation will be analyzed. It is hoped or believed that the experience and knowledge obtained in this study will contribute to the fluid mechanical aspects of the overall research in hemodynamics and clinical application of cardiac assist devices. The proposed research project will be concentrated on or limited to laminar flows with transient vortices. Although limited in scope, some in-vitro and in-vivo experiments will be conducted for the needed data in computational analyses.